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On the Order of Nilpotent Multipliers of
Finite p-Groups∗
Behrooz Mashayekhy and Mahboobeh Alizadeh Sanati
Department of Mathematics, Ferdowsi University of Mashhad,
P.O.Box 1159-91775, Mashhad, Iran
Institute for Studies in Theoretical Physics and Mathematics,
Tehran, Iran
Abstract
Let G be a finite p-group of order pn. YA. G. Berkovich (Journal
of Algebra 144, 269-272 (1991)) proved that G is elementary abelian
p-group if and only if the order of its Schur multiplier, M(G), is at
the maximum case. In this paper, first we find the upper bound
pχc+1(n) for the order the c-nilpotent multiplier of G, M (c)(G), where
χc+1(i) is the number of basic commutators of weight c + 1 on i let-
ters. Second, we obtain the structure of G, in abelian case, where
|M (c)(G)| = pχc+1(n−t), for all 0 ≤ t ≤ n − 1. Finally, by putting
a condition on the kernel of the left natural map of the generalized
Stallings-Stammbach five term exact sequence, we show that an arbi-
trary finite p-group with the c-nilpotent multiplier of maximum order
is an elementary abelian p-group.
∗This research was in part supported by a grant from IPM.
1
Key words and phrases: Nilpotent multipliers, finite p-group, elemen-
tary abelian p-group.
A.M.S.Classification 2000 : 20D15, 20E10, 20K01.
1.Introduction and Motivation
Let G be a group with a free presentation
1 −→ R −→ F −→ G −→ 1 ,
where F is a free group and R a normal subgroup of F such that the above
sequence is exact. This paper focuses on the abelian group
M (c)(G) =R ∩ γc+1(F )
[R,c F ],
where γc+1(F ) is the (c+ 1)st term of the lower central series of F and
[R,1 F ] = [R,F ] ,
[R,c F ] = [[R,c−1 F ], F ] .
R. Baer [1] showed that M (c)(G) is independent of the choice of the free
presentation of G. The group M (1)(G) = M(G) is the much studied Schur
multiplier of G. We shall call M (c)(G) the c-nilpotent multiplier of G. In
fact, M (c)(G) is the well-known notion the Baer invariant of G with respect
to the variety of nilpotent groups of class at most c, Nc. We also denote
M (c)(G) by NcM(G) (See [11]).
In 1907, I. Schur [16], using representation method, showed that if G
is the direct product of A and B, then the Schur multiplier of G has the
following form:
M(G) = M(A× B) ∼= M(A)⊕M(B)⊕ (Aab ⊗Bab).
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A useful consequence of the above fact is to obtain an explicit formula for
the structure of the Schur multiplier of a finite abelian group, in particular
for an elementary abelian p-group.
In 1956, J. A. Green [6] proved that ifG is a finite p-group of order pn, then
M(G), the Schur multiplier of G, is of order pm(G), where m(G) ≤ 12n(n−1).
In 1991, YA. G. Berkovich [3], using the above facts and some properties
of covering groups proved that a finite p-group G of order pn is elementary
abelian p-group if and only if the order of M(G) is pm(G), where m(G) =12n(n− 1).
Now, in this paper, we are going to concentrate on the Berkovich’s result
for the c-nilpotent multiplier of a finite p-group.
2.Notation and Prelimineries
In this section, we are going to recall some notions which are used in
future.
Definition 2.1 [7]
The notion of basic commutators on letters x1, x2, . . . , xn, . . . , are defined
as follows:
(i) The letters x1, x2, . . . , xn, . . . are basic commutators of weight one, ordered
by setting xi < xj if i < j.
(ii) If basic commutators ci of weight wt(ci) < k are defined and ordered,
then define basic commutators of weight k by the following rules:
[ci, cj] is a basic commutator of weight k if
1. wt(ci) + wt(cj) = k ,
2. ci > cj ,
3. if ci = [cs, ct], then cj ≥ ct .
Then continue the order by setting c > ci whenever wt(c) > wt(ci) and fixing
any order amongst those of weight k and finally numbering them in order.
Theorem 2.2 (P.Hall [7,8])
3
Let F =< x1, x2, . . . , xd > be a free group, then
γn(F )
γn+i(F ), 1 ≤ i ≤ n
is the free abelian group freely generated by the basic commutators of weights
n, n+ 1, . . . , n+ i− 1 on the letters {x1, . . . , xd}.
Theorem 2.3 (Witt Formula [7])
The number of basic commutators of weight n on d generators is given
by the following formula:
χn(d) =1
n
∑
m|n
µ(m)dn/m
where µ(m) is the Mobious function, and defined to be
µ(m) =
1 if m = 1,
0 if m = pα1
1 . . . pαk
k ∃αi > 1,
(−1)s if m = p1 . . . ps,
Definition 2.4
Let G be a group with a free presentation
1 −→ R −→ F −→ G −→ 1 ,
where F is a free group. Then the Baer invariant of G with respect to the
variety V, denoted by VM(G) is defined to be
VM(G) =R ∩ V (F )
[RV ∗F ].
One can see that the Baer invariant of the group G is always abelian and
independent of the choice of the presentation of G. ( See C. R. Leedham-
Green and S. McKay [11], from where we have taken our notation and H.
Neumann [15] for the notion of varieties of groups.)
4
In particular, if V is the variety of abelian groups, A, then the Baer
invariant of the group G will be
R ∩ F ′
[R,F ],
which is isomorphic to the Schur multiplier of G, denoted by M(G), where,
in finite case, is the second cohomology group of G, H2(G,C∗). Also if V is
the variety of nilpotent groups of class at most c ≥ 1, Nc, then the Baer
invariant of the group G will be
NcM(G) =R ∩ γc+1(F )
[R, cF ].
We also call it the c-nilpotent multiplier of G.
Definition 2.5
A variety V is said to be a Schur-Baer variety if for any group G for which
the marginal factor group, G/V ∗(G), is finite, then the verbal subgroup,V (G),
is also finite and |V (G)| divides a power of |G/V ∗(G)|. I. Schur in [16] proved
that the variety of abelian groups, A, is a Schur-Baer variety. Also R. Baer in
[2] proved that the variety defined by some outer commutator words has the
above property. The following theorem tells us a very important property of
Schur-Baer varieties.
Theorem 2.6(C. R. Leedham-Green and S. McKay [11])
The following conditions on the variety V are equivalent:
(i) V is a Schur-Baer variety.
(ii) For every finite group G, its Baer invariant, VM(G), is of order dividing
a power of |G| .
Theorem 2.7 (I. Schur [10, 16])
Let
G ∼= Zn1⊕ Zn2
⊕ . . .⊕ Znk
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be a finite abelian group, where ni+1|ni for all 1 ≤ i ≤ k−1 and k ≥ 2. Then
M(G) ∼= Zn2⊕ Z(2)
n3⊕ . . .⊕ Z(k−1)
nk,
where Z(m)n denote the direct sum of m copies of Zn .
Theorem 2.8 (M. R. Jones [9, 10])
Let G be a finite p-group of order pn and let G admit a presentation with
d generators. Then
p1/2d(d−1) ≤ |M(G)||G′| ≤ p1/2n(n−1) .
Theorem 2.9 (J. A. Green [6])
Let G be a finite p-group of order pn. Then the order of M(G) is at most
p1/2n(n−1) .
Theorem 2.10 (YA. G. Berkovich [3])
Let G be a finite p-group of order pn. Then |M(G)| = p1/2n(n−1) if and
only if G is an elementary abelian p-group.
Definition 2.11 [12]
By a pre-crossed module we mean a pair of groups (A,G) and homomor-
phism α : A −→ G, together with an action of G on A which satisfies the
condition
α(ag) = g−1α(a)g for all g ∈ G and a ∈ A .
A pre-crossed module for which, in addition, we have
aα(b) = b−1ab for all a, b ∈ A
is called a crossed module.
Let A be a G-precrossed module, then the non-abelian tensor product
A⊗G is defined to be the group with generators
{< a, x > |a ∈ A, x ∈ G}
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subject to the relations
< a1a2, x >=< aα(a1)2 , xα(a1) >< a1, x >
< a, x1x2 >=< a, x1 >< ax1, xx1
2 > .
Theorem 2.12 (J. Stallings and U. Stammbach [4])
Every extension 1 → N → G → Q → 1 determines an exact sequence
M(G) −→ M(Q) −→ N/[N,G] −→ G/G′ −→ Q/Q′ −→ 1 .
3. The Main Results
In order to present some main results we need the following theorems.
Theorem 3.1 (B. Mashayekhy and M. R. R. Moghaddam [13])
Let G ∼= Zn1⊕Zn2
⊕ . . .⊕Znkbe a finite abelian group, where ni+1|ni for
all 1 ≤ i ≤ k− 1 and k ≥ 2. Then, for all c ≥ 1, the c-nilpotent multiplier of
G is
NcM(G) = M (c)(G) ∼= Z(b2)n2
⊕ Z(b3−b2)n3
⊕ . . .⊕ Z(bk−bk−1)nk
,
where bi is the number of basic commutators of weight c + 1 on i letters.
Note that Z(m)n denotes the direct sum of m copies of Zn.
Theorem 3.2 (M. R. R. Moghaddam [14])
Let V be a variety of polynilpotent groups and let G be a p-group of order
pn, where p is any prime number. Then
|VM(G)||V (G)| ≤ |VM(Z(n)p )| .
Now, we can get an upper bound for the order of a c-nilpotent multiplier
of a finite p-group as follows.
Corollary 3.3
Let G be a p-group of order pn. Then the order of M (c)(G) is at most
pχc+1(n), where χc+1(n) is the number of basic commutators of weight c + 1
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on n generators.
Proof.
Since Z(n)p
∼= Zp ⊕ Zp ⊕ . . .⊕ Zp (n-copies), by Theorem 3.1 we have
M (c)(Z(n)p ) ∼= Z(b2)
p ⊕ Z(b3−b2)p ⊕ . . .⊕ Z(bk−bk−1)
p .
Therefore |M (c)(Z(n)p )| = pb2+(b3−b2)+...+(bn−bn−1) = pbn = pχc+1(n).
Now, by Theorem 3.2 we have
|M (c)(G)| ≤ |M (c)(G)||γc+1(G)| ≤ |M (c)(Z(n)p )| = pχc+1(n) .✷
Note that Corollary 3.3 is a vast generalization of Green’s result (Corol-
lary 2.9), since χ2(n) = 1/2∑
m|2 µ(m)n2/m = 1/2(µ(1)n2+µ(2)n1) = 1/2(n2−
n) = 1/2n(n− 1).
Now in the following we are going to present a vast generalization of the
Berkovich’s result (Theorem 2.10) in abelian case.
Theorem 3.4
Let G be an abelian p-group of order pn. Then for all c ≥ 1 and all
0 ≤ t ≤ n− 1
|M (c)(G)| = pχc+1(n−t) if and only if G ∼= Zpt+1 ⊕ Zp ⊕ . . .⊕ Zp︸ ︷︷ ︸
n−t−1−copies
.
Proof.
LetG ∼= Zpt+1⊕Zp ⊕ . . .⊕ Zp︸ ︷︷ ︸
n−t−1−copies
. Then, by Theorem 3.1, we haveM (c)(G) ∼=
Z(b2)p ⊕ Z(b3−b2)
p ⊕ . . .⊕ Z(bn−t−bn−t−1)p . Hence
|M (c)(G)| = pb2+(b3−b2)+...+(bn−t−bn−t−1) = pbn−t = pχc+1(n−t) .
Conversely, suppose that |M (c)(G)| = pχc+1(n−t) . Since G is an abelian p-
group of order pn, so by the fundamental theorem of finitely generated abelian
groups we have
G ∼= Zpα1 ⊕ Zpα2 ⊕ . . .⊕ Zpαk ,
8
where α1 ≥ α2 ≥ . . . ≥ αk and α1 + α2 + . . . + αk = n . By Theorem 3.1
M (c)(G) ∼= Z(b2)pα2 ⊕ Z
(b3−b2)pα3 ⊕ . . .⊕ Z
(bk−bk−1)pαk
. Thus
|M (c)(G)| = pα2b2+α3(b3−b2)+...+αk(bk−bk−1) .
On the other hand, by hypothesis |M (c)(G)| = pχc+1(n−t) = pbn−t . Therefore
we have bn−t = α2b2 + α3(b3 − b2) + . . . + αk(bk − bk−1) . Since α2 ≥ αi for
all 2 ≤ i ≤ k, we have
bn−t ≤ α2b2 + α2(b3 − b2) + . . .+ α2(bk − bk−1) = α2bk (I) .
Also αk ≤ αi for all 2 ≤ i ≤ k, so we have
bn−t ≥ αkb2 + αk(b3 − b2) + . . .+ αk(bk − bk−1) = αkbk (II) .
If n = 1 then t = 0. So we have G ∼= Zp, |M (c)(G)| = 1 = pb1. Hence the
result holds .
If n = 2 then there are two cases t = 0 or t = 1. If t = 0, then |M (c)(G)| = pb2
and so G ∼= Zp ⊕ Zp. If t = 1 then |M (c)(G)| = pb1 = 1 and G ∼= Zp2 . Hence
the result holds. Therefore we can assume that n ≥ 3.
Now, we claim that k = n− t. In order to prove the claim, consider the
following two cases.
Case one: If k > n− t, then clearly we have the inequalities αkbk ≥ bk >
bn−t which contradicts to inequality (II).
Case two: If k < n − t. Put k = n − t − j for some 1 ≤ j < n − t − 1.
Clearly α2 is maximum when α1 = α2 and α3 = . . . = αk = 1 . So in this
case 2α2 + k − 2 = n. Hence we have α2 ≤ (t+ j + 2)/2 . So by (I) we have
the following inequality:
bn−t ≤t+ j + 2
2bn−t−j .
Now, we need the following inequality which is proved by induction on n ≥ 3.
t + j + 2 ≤ 2(n− (t+ 1)) . . . (n− (t+ j)) (III) ,
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where 0 ≤ t ≤ n− 1and1 ≤ j < n− t− 1. If n = 3, then t = 0 and j = 1. So
0 + 1 + 2 ≤ 2(3− 1) = 4 .
Now, let (III) holds for n with the above condition and we investigate
the following inequality:
t+ j + 2 ≤ 2(n− t) . . . (n+ 1− (t + j)), 0 ≤ t < n, 1 ≤ j < n− t .
If 0 ≤ t < n−1 and 1 ≤ j < n− t−1, then by induction hypothesis we have
t+ j + 2 ≤ 2(n− (t + 1)) . . . (n− (t+ j)) ≤ 2(n− t) . . . (n+ 1− (t + j)) .
If 0 ≤ t < n − 1 and j = n − t − 1, then k = 1 and so G ∼= Zpn. Hence
|M (c)(G)| = 1 = pbn−t . Thus t = n− 1 and j = 0 which is a contradiction to
j ≥ 1.
Also, if t = n − 1 and 1 ≤ j < n − t, then 1 ≤ j < n − (n − 1) = 1,
which is a contradiction. Hence the last two cases do not happen. Thus, the
following inequalities hold by (III):
bn−t ≤t+ j + 2
2bn−t−j ≤ (n− (t+ 1)) . . . (n− (t+ j))bn−t−j .
By a routine combinatorial disscusion and considering the definition of basic
commutators we can see that ibi < bi+1 . (Note that we can choose i letters
from i+ 1 letters in i cases.) Therefore we have
bn−t ≤ (n−(t+1)) . . . (n−(t+j))bn−t−j < (n−(t+1)) . . . (n−(t+j−1))bn−(t+j−1)
< . . . < (n− (t+ 1))bn−(t+1) < bn−t .
which is a contradiction. Hence k = n − t and our claim is true. So we can
consider
G ∼= Zpα1 ⊕ Zpα2 ⊕ . . .⊕ Zpαn−t .
10
Now we show that α1 = t+1 and α2 = . . . = αn−t = 1 . We have |M (c)(G)| =
pα2b2+α3(b3−b2)+...+αn−t(bn−t−bn−t−1) . By (II) we have αn−tbn−t ≤ bn−t and so
αn−t = 1. Also, αn−t−1 ≤ αi for 2 ≤ i ≤ n− t− 1, thus
αn−t−1b2+αn−t−1(b3−b2)+. . .+αn−t−1(bn−t−1−bn−t−2)+(bn−t−bn−t−1) ≤ bn−t .
Hence we can conclude that αn−t−1 = 1. Similarly, one can see that α2 =
. . . = αn−t−1 = 1. Since α1+. . .+αn−t = n, we have α1 = t+1, as required. ✷
Now, the following corollary is a generalization of Berkovich’s result (The-
orem 2.10).
Corollary 3.5
Let G be an abelian p-group of order pn. Then |M (c)(G)| = pχc+1(n) if
and only if G is the elementary abelian p-group.
Now, in order to deal with non-abelian case, we need the following two
important results.
Theorem 3.6 (A. Frohlich [5])
Let 1 → N → G → Q → 1 be a V-central extension, where V is any
variety of groups (this means that the above sequence is exact and N is
contained in the marginal subgroup of G, V ∗(G)). Then the following five-
term exact sequence exists:
VM(G)θ
−→ VM(Q) −→ N −→ G/V (G) −→ Q/V (Q) −→ 1 .
Theorem 3.7 (A. S.-T. Lue [12])
Let 1 → N → G → Q → 1 be an Nc-central extension. Then the
following sequence is exact:
N ⊗G/γc+1(G)⊗ . . .⊗G/γc+1(G)︸ ︷︷ ︸
c−copies
−→ NcM(G)θ
−→ NcM(Q)
11
−→ N −→ G/γc+1(G) −→ Q/γc+1(Q) −→ 1 ,
where the above tensor product is the non-abelian tensor product. Moreover
kerθ =[S,c F ]
[R,c F ],
where G = F/R , Q = F/S are free presentations for G and Q, respectively.
Theorem 3.8
Let G be a finite p-group of order pn. If |M (c)(G)| = pχc+1(n), then
(i) There is an epimorphism M (c)(G)θ
−→ M (c)(G/G′) which is obtain from
the Frohlich’s sequence.
(ii) If kerθ = 1, then G is an elementary abelian p-group.
Proof.
(i) By Theorem 3.2 and Corollary 3.3
|M (c)(G)||γc+1(G)| ≤ |M (c)(Z(n)p )| = pχc+1(n) .
Since |M (c)(G)| = pχc+1(n), we have |γc+1(G)| = 1.
Now, set N = G′ and consider the exact sequence
1 −→ G′ −→ G −→ G/G′ −→ 1 .
Since γc+1(G) = 1, the above exact sequence is an Nc-central extension.
Therefore by Theorem 3.6 we have the following exact sequence:
M (c)(G)θ
−→ M (c)(G/G′)β
−→ G′ α−→ G −→ G/G′ −→ 1 .
Clearly α is monomorphism and so Imβ = 1. This means that θ is epimor-
phism.
(ii) Let kerθ = 1, then
|M (c)(G/G′)| = |M (c)(G)| = pχc+1(n) .
Since |G| = pn, so |G/G′| ≤ pn. Hence by Corollary 3.3 we have |G/G′| = pn.
This means that G′ = 1 and so G is abelian. Now by Corollary 3.5 G is an
elementary abelian p-group. ✷
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References
[1] Baer, R. Representations of Groups as Quotient Groups. I,II,III, Trans.
Amer. Math. Soc. 1945, 58, 295-419.
[2] Baer, R. Endlichkeitskriterien fur Kommutatorgruppen. Math. Ann.
1952, 124, 161-177.
[3] Berkovich, YA. G. On the Order of the Commutator Subgroup and the
Schur Multiplier of a finite p-Group. Journal of Algebra 1991, 144, 269-
272.
[4] Byle, F. R.; Tappe, J. Group extension, Representation and the Schur
Multiplicator, Lecture Note in Mathematics No. 958. Springer-Verlag:
Berlin, 1974.
[5] Frohlich, A. Baer Invariants of Algebras. Trans. Amer. Soc. 1963, 109,
221-244.
[6] Green, J. A. On the Number of Automorphism of a Finite Group. Proc.
Roy. Soc. A 1956, 237, 574-581.
[7] Hall, M. The Theory of Groups, MacMillan Company: New York, 1959.
[8] Hall, P. Nilpotent Groups. Cand. Math. Congress, Univ. of Alberta
1957. Queen Mary College Math. Notes, London, 1970.
[9] Jones, M. R. Numerical Results on Multiplicators of Finite Groups.
Ph.D. Thesis, University of Wales, 1973.
[10] Karpilovsky, G. The Schur Multiplier, London Math. Soc. Monographs,
New Series no. 2, 1987.
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[11] Leedham-Green, C. R.; McKay, S. Baer-invariant,Isologism, Varietal
Laws and Homology. Acta Math. 1976, 137, 99-150.
[12] Lue, A. S.-T. The Ganea map for Nilpotent Groups. J. London Math.
Soc. 1976, 14, 309-312.
[13] Mashayekhy, B.; Moghaddam, M. R. R. Higher Schur Multiplicator of
a Finite Abelian Group. Algebra Colloquium 1997, 4(3), 317-322.
[14] Moghaddam, M. R. R. On the Schur-Baer Property. J. Austral. Math.
Soc. (Series A) 1981, 31, 343-361.
[15] Neumann, H. Varieties of Groups, Springer-Verlag: Berlin, 1967.
[16] Schur, I. Untersuchungen uber die Darstellung der Endlichen Gruppen
durch Gebrochene Lineare Substitutionen. J. Reine Angew. Math. 1907,
132, 85-137.
E-mail: [email protected]
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